- Research Article
- Open Access
Second-Order Boundary Value Problem with Integral Boundary Conditions
© Mouffak Benchohra et al. 2011
- Received: 28 May 2010
- Accepted: 1 October 2010
- Published: 13 October 2010
The nonlinear alternative of the Leray Schauder type and the Banach contraction principle are used to investigate the existence of solutions for second-order differential equations with integral boundary conditions. The compactness of solutions set is also investigated.
- Differential Equation
- Banach Space
- Unique Solution
- Ordinary Differential Equation
- Functional Equation
Boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint, and nonlocal boundary value problems as special cases. For boundary value problems with integral boundary conditions and comments on their importance, we refer the reader to the papers [1–9] and the references therein. Moreover, boundary value problems with integral boundary conditions have been studied by a number of authors, for example [10–14]. The goal of this paper is to give existence and uniqueness results for the problem (1.1). Our approach here is based on the Banach contraction principle and the Leray-Schauder alternative .
In this section, we introduce notations, definitions, and preliminary facts that will be used in the remainder of this paper. Let be the space of differentiable functions whose first derivative, , is absolutely continuous.
Our first result reads
then the BVP (1.1) has a unique solution.
We now present an existence result for problem (1.1).
Suppose that hypotheses
Transform the BVP (1.1) into a fixed-point problem. Consider the operator as defined in Theorem 3.3. We will show that satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof will be given in several steps.
As the right-hand side of the above inequality tends to zero. Then is equicontinuous. As a consequence of Steps 1 to 3 together with the Arzela-Ascoli theorem we can conclude that is completely continuous.
Step 4 (A priori bounds on solutions).
and consider the operator From the choice of , there is no such that for some As a consequence of the nonlinear alternative of Leray-Schauder type , we deduce that has a fixed point in which is a solution of the problem (1.1).
We present some examples to illustrate the applicability of our results.
The authors are grateful to the referees for their remarks.
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